AP Calculus AB Summer Packet 2016 - Fall River Public … Calculus AB Summer Assignment 2016 (1).… · B.M.C. Durfee High School Mathematics Department Page 1 . AP CALCULUS AB

B.M.C. Durfee High School Mathematics Department Page 1

AP CALCULUS AB

Summer Assignment 2016

AP Calculus AB Summer Packet 2016


Welcome to AP Calculus Part AB. This may be the toughest mathematics class in your mathematical careers, but the benefit you will receive by having this experience in high school is immense. Because of the unique nature of this class, it is very important that you are ready to start working on the first day. We will NOT spending time to review the material in this packet. The reason for this is to make sure we have adequate time complete all of the required material for this course well before the AP exam. In AP Calculus, there are extremely high expectations of students taking the course. We expect a certain level of independence to be demonstrated by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness is through this summer work packet which will help you maintain/improve your skills. This packet is a requirement for those entering AP Calculus AB and is due on the first day of class. It will be counted as an assessment grade. The entire packet will be graded for completeness and correctness. There will also be an in-class assessment on the material from this packet during the first week of school. There are certain math skills that have been taught to you over the previous years that are necessary to be successful in calculus. If you do not fully understand the topics in this packet, it is possible that you will get calculus problems wrong in the future not because you do not understand the calculus concept, but because you do not understand the algebra or trigonometry behind it. Don’t fake your way through any of these problems because you will need to understand the underlying concepts to be successful in this course. You may work with someone else while you do this, but copying will not be tolerated. Also, don’t wait until the last minute to do everything in the packet because you may run out of time and rush through them. Likewise, do not do all the problems right at the beginning of summer and completely forget how to do all of them by the time school starts again. For this packet you must show all of your work for the multiple choice questions on a separate sheet of paper and fill in your answers on the attached answer sheet. Also, do not rely on a calculator to do all of the work for you. Half of the AP exam does not allow any calculator at all. This packet will be worth an assessment grade. Enjoy your summer and don’t forget about the packet. September will be here before you know it! If you lose your packet, you will be able to access the packets on-line at the school website or a copy can be picked up in the Registrar’s office. Best of luck with the assignment! I look forward to seeing you in September! Mrs. Brogan



Required Saturday Sessions

Please be advised that you are required to attend the two Saturday sessions and mock exam day. The dates for the Saturday sessions are below:

Session #1: January 7, 2017 at Durfee;

Session #2: March 25, 2017 at Middleborough

The date for the mock exam will be announced the first day of class. It has been tentatively scheduled for April 1, 2017.

Failure to attend any Saturday Session or the mock exam will result in the students being required to take the final exam as well as an additional assignment.

Calculator Required

A graphing calculator is required for this class. You must have a calculator on the first day of class. We will be using a TI-84 Plus C Silver Edition for examples in class, so this is the recommended calculator. This calculator is normally on sale at Target and WalMart in August, so keep your eyes open for sales. The regular price for this calculator is approximately $115.

If you choose to purchase a different graphing calculator, it is your responsibility to figure out how to use it. All graphing calculators work a little differently and the teacher may not be familiar with the calculator you purchase.

The graphing calculator is required on half of the AP exam, therefore it is necessary for you to become comfortable using it prior to the exam. It will be used on a regular basis in class and on homework assignments. Make sure you have the most recent OS version.



Directions: Identify the choice that best completes the statement or answers the question. Show your work on a separate sheet of paper and fill in your answers on the bubble sheet that has been provided in this packet.

1. Simplify:

a. c.

b.

d.

2. Simplify:

a.

c.

b.

d.

3. Simplify: +

a.

c.

b. d.

4. Simplify:

a.

c.

b. d.



5. Simplify: Assume that no denominator is equal to 0.

a.

c.

b. d.

6. Find the inverse of the given function:

a. c.

b. d.

7. Simplify .

a.

c.

b. d.

8. Simplify .

a. c.

b. d.

9. Find the domain of the function: 𝒇𝒇(𝒙𝒙) = 𝒙𝒙𝟐𝟐−𝟗𝟗𝒙𝒙+𝟐𝟐

𝒙𝒙𝟐𝟐−𝟒𝟒𝒙𝒙+𝟑𝟑

a. All real numbers b. All real numbers

c. All real numbers d. All real numbers

x ≠ 3 1,

x ≠ – , –5 4

x ≠ – , –3 1

x ≠ 5 4,



10. Factor the expression and use the fundamental identities to simplify.

a. b.

c. 1 d.

11. Find the expression that completes the identity:

a. b. 0

c. d.

12. Which equation represents a line through (-1, 1) with a slope of 𝟐𝟐

𝟑𝟑?

a. 𝑦𝑦 − 1 = 2

3(𝑥𝑥 + 1)

b. 𝑦𝑦 + 1 = 2

3(𝑥𝑥 − 1)

c. 𝑦𝑦 − 1 = 23

(𝑥𝑥 − 1)

d. 𝑦𝑦 + 1 = 23

(𝑥𝑥 + 1)

13. Which of the following equations is shown in the graph below?

a. 𝑦𝑦 + 2 = −12

(𝑥𝑥 + 2)

b. 𝑦𝑦 + 3 = −12

(𝑥𝑥 + 6)

c. 𝑦𝑦 − 3 = −12

(𝑥𝑥 − 6)

d. 𝑦𝑦 − 2 = −12

(𝑥𝑥 − 2)

cos sec cos2 2 2x x x−

cos cot2 2x x

cos2 x sin2 x

1 cos sin

sin 1 cos

−+

−=

uu

uu

2csc u 2sin u

2 cos + u



14. Write 𝟏𝟏𝟐𝟐𝟏𝟏𝟑𝟑 ∙ 𝟒𝟒𝟒𝟒

𝟏𝟏𝟑𝟑 ∙ 𝟒𝟒𝟓𝟓

𝟏𝟏𝟑𝟑 in simplest form.

a. √27,000

b. 30

c. 10713

d. 27,000

15. What is (−𝟑𝟑𝟐𝟐𝒙𝒙𝟏𝟏𝟓𝟓𝒚𝒚𝟑𝟑𝟒𝟒)−𝟏𝟏𝟒𝟒 in simplest form?

a. 2𝑥𝑥2𝑦𝑦7

b. − 2𝑥𝑥2𝑦𝑦7

c. − 12𝑥𝑥2𝑦𝑦7

d. 2𝑥𝑥2𝑦𝑦7


Short Answer Section: All work must be done on a separate sheet of paper. You must show work!

17. Factor completely: 𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 − 𝟖𝟖𝒙𝒙 + 𝟏𝟏

18. Factor completely: 𝟑𝟑𝟏𝟏𝒙𝒙𝟐𝟐 − 𝟏𝟏𝟓𝟓𝟓𝟓𝒚𝒚𝟐𝟐

19. Factor completely: 𝟖𝟖𝒙𝒙𝟑𝟑 − 𝟐𝟐𝟐𝟐

20. Factor completely: 𝟏𝟏𝟓𝟓𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟑𝟑𝒙𝒙 − 𝟑𝟑

21. Factor completely and simplify: (𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟒𝟒)𝟐𝟐(𝟑𝟑𝒙𝒙𝟐𝟐 + 𝟒𝟒𝒙𝒙 − 𝟏𝟏) + (𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟒𝟒)𝟑𝟑(𝒙𝒙 − 𝟒𝟒)

22. Factor completely. There should be no fractional or negative exponents in your answer.

a. 2√𝑥𝑥 − 6𝑥𝑥32 b. 𝑒𝑒−𝑥𝑥 − 𝑥𝑥𝑒𝑒−𝑥𝑥 + 2𝑥𝑥2𝑒𝑒−𝑥𝑥 c. 2𝑥𝑥

54 + 𝑥𝑥

34 − 15𝑥𝑥

14

23. Solve for y’ and write your answer in simplest form.

a. 𝑥𝑥𝑦𝑦′ + 𝑦𝑦 = 1 + 𝑦𝑦′ b. 3𝑦𝑦2𝑦𝑦′ + 2𝑦𝑦𝑦𝑦′ = 5𝑦𝑦′ + 2𝑥𝑥 c. 3𝑥𝑥2𝑦𝑦𝑦𝑦′ + 2𝑥𝑥𝑦𝑦2 = 2𝑦𝑦𝑦𝑦′

16.





24. Determine the exact value (without a calculator). No Decimal Answers!!!

a. cos 0 b. sin𝜋𝜋 c. tan 𝜋𝜋2

d. cos 5𝜋𝜋6

e. sin 4𝜋𝜋3

f. tan 3𝜋𝜋4

g. sec 5𝜋𝜋3

h. csc 7𝜋𝜋4

i. cot 7𝜋𝜋6

j. sin(−3𝜋𝜋2

) k. cos 5𝜋𝜋 g. tan(−11𝜋𝜋6

)

25. Solve each trigonometric equation below. Find the exact answer on the interval 𝟓𝟓 ≤ 𝒙𝒙 ≤ 𝟐𝟐𝟐𝟐. No

Decimal Answers!!!

a. 4 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑥𝑥 − 3 = 0

b. √2 sin(2𝑥𝑥) = 1

c. 3 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑥𝑥 − 1 = 0

d. 𝑐𝑐𝑐𝑐𝑐𝑐3𝑥𝑥 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑥𝑥

e. sin𝑥𝑥 − 2 sin𝑥𝑥 cos 𝑥𝑥 = 0

f. 2𝑐𝑐𝑠𝑠𝑠𝑠2𝑥𝑥 − 𝑐𝑐𝑠𝑠𝑠𝑠𝑥𝑥 − 3 = 0

g. 𝑐𝑐𝑐𝑐𝑐𝑐2𝑥𝑥 = 1 − 𝑐𝑐𝑠𝑠𝑠𝑠𝑥𝑥

h. √3 + tan(2𝑥𝑥) = 0

i. cos 𝑥𝑥 = sin𝑥𝑥

j. sin(3𝑥𝑥) = −1



26. Let 𝒇𝒇(𝒙𝒙) = 𝟐𝟐𝒙𝒙 + 𝟏𝟏 and 𝒈𝒈(𝒙𝒙) = 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟏𝟏. Find each.

a. 𝑓𝑓(2)

b. 𝑔𝑔(−3)

c. 𝑓𝑓(𝑔𝑔(4))

d. 𝑔𝑔(𝑓𝑓(𝑥𝑥))

e. 𝑔𝑔�𝑓𝑓(𝑥𝑥 − 1)�

27. Find the vertical and horizontal asymptotes if possible.

a. 𝑓𝑓(𝑥𝑥) = 4−𝑥𝑥𝑥𝑥2−16

b. 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥2+𝑥𝑥−3𝑥𝑥2+𝑥𝑥−2

28. Determine the equation of a line passing through the point (5, -3) with a slope of 4. Write the equation

in point-slope form.

29. Find the inverse of the function 𝒚𝒚 = √𝟒𝟒 − 𝒙𝒙 + 𝟏𝟏



30. Graph each of the following piecewise functions. Determine any points of discontinuity.

a. 𝑓𝑓(𝑥𝑥) = �𝑥𝑥 + 5; 𝑥𝑥 ≤ 2−4; 𝑥𝑥 > 2

b. ℎ(𝑥𝑥) = �𝑥𝑥 − 1; 𝑥𝑥 ≤ −2

2𝑥𝑥 − 1; −2 < 𝑥𝑥 ≤ 4−3𝑥𝑥 + 8; 𝑥𝑥 > 4



31. Solve each logarithmic or exponential equation below. Give exact answers.

a. log20(8 − 2𝑥𝑥) = log20(−3𝑥𝑥 + 10)

b. 5 ln(𝑏𝑏 − 9) = 20

c. 3 log4(4𝑠𝑠 − 5) + 4 = 7

d. log(𝑥𝑥 + 21) + log 𝑥𝑥 = 2

e. 52𝑥𝑥 = 20

f. 4 − 2𝑒𝑒𝑥𝑥+1 = −12

Graphing Calculator Portion: Use your graphing calculator to answer the following questions. When

necessary, round your answers to three decimal places.

32. Find the point(s) of intersection of the graphs for the given equations.

a. � 𝑥𝑥 + 𝑦𝑦 = 84𝑥𝑥 − 𝑦𝑦 = 7

b. �𝑥𝑥2 + 𝑦𝑦 = 6𝑥𝑥 + 𝑦𝑦 = 4

33. Find the relative maximum and the relative minimum for the function 𝒇𝒇(𝒙𝒙) = 𝒙𝒙𝟑𝟑 + 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟗𝟗𝒙𝒙 − 𝟏𝟏.

34. Find the zeros for the function 𝒇𝒇(𝒙𝒙) = 𝒙𝒙𝟑𝟑 + 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟗𝟗𝒙𝒙 − 𝟏𝟏.



35. Using the calculator, create a table of values for the equation 𝒇𝒇(𝒙𝒙) = 𝒙𝒙√𝟏𝟏𝟏𝟏 − 𝒙𝒙𝟐𝟐. Copy and fill in the

table below:

x f(x)

-3

-2

-1

0

1

2

3



Use the graph below to answer questions 36-51.

36. =−−→

)(lim3

xfx

37. =+→

)(lim4

xfx

38. =+−→

)(lim3

xfx

39. =→

)(lim4

xfx

40. =−→

)(lim3

xfx

41. =−→

)(lim6

xfx

42. =+−→

)(lim6

xfx

43. =−→

)(lim2

xfx

44. =→

)(lim6

xfx

45. =+→

)(lim2

xfx

46. =→

)(lim2

xfx

47. =−∞→

)(lim xfx

48. =−→

)(lim4

xfx

49. =∞→

)(lim xfx

50. =)2(f 51. =)3(f



52. Find the 1

1lim 21 −−

→ xx

x 53. Find the )432(lim 2

5+−

→xx

x

54. Find the x

xxx 35

12lim23

2 −−+

−→ 55. Find the

96lim 2

2

3 −−+

−→ xxx

x

56. Find the 28lim

3

2 ++

−→ xx

x



Free Response Section

Part I:

Let f be a function defined on the closed interval [0,7]. The graph of f, consisting of four line segments, is shown above.

57. Write the piecewise function for f(x).

58. Find the values of x for which 𝑓𝑓(𝑥𝑥) = 1. Justify your answer.

59. For what interval(s) if x is the rate of change of 𝑓𝑓(𝑥𝑥) positive? Justify your answer.

60. For what interval(s) if x is the rate of change of 𝑓𝑓(𝑥𝑥) negative? Justify your answer.

61. For what interval(s) if x does f have a rate of change of 2? Justify your answer.

62. On which interval of x is the rate of change of f the greatest? Justify your answer.

63. For what x value does 𝑓𝑓(𝑥𝑥) reach a maximum value? Justify your answer.



Part II

The graph of f(x) is shown above.

64. Write an equation of the secant line connecting the points (0, -1) and (2, -1). Draw all of the possible tangent lines of f(x) that is parallel to the secant line.

65. Find any local extrema and justify why these points are local extrema.

66. Find any absolute extrema.

67. Estimate the intervals for which f(x) is concave up and concave down.

68. Estimate the inflection point(s) on f(x)



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AP Calculus AB Summer Packet 2016 - Fall River Public … Calculus AB Summer Assignment 2016 (1).… · B.M.C. Durfee High School Mathematics Department Page 1 . AP CALCULUS AB